Another prominent class of methods, ellipsoidal techniques
[1]}, solve the more general problem of feedback
synthesis for linear time-varying plants with two competing
inputs. When applied to the system REF , these methods
yield a set-valued function \(E\)  defined on the interval
\({t_0,t_f}\)  whose value at any time is a finite intersection of
ellipsoids containing the reachable set at that time as a
subset. While arbitrarily precise over-approximations are obtained
when a sufficient amount of ellipsoids is computed, the approach
suffers from two shortcomings. Firstly, the ellipsoids result from
numerically solving linear-quadratic optimal control problems derived
from REF , yet numerical errors incurred in the course
of the solution are not taken into account. Hence, mere approximations
rather than over-approximations might actually be computed.
Secondly, approximations of reachable tubes are obtained only
implicitly, as the union over \(t \in {t_0,t_f}\)  of \(E(t)\) , and
so they are disjoint from an unsafe region \(R\) 
if and only if the graph of the set-valued map \(E\)  is disjoint from
the set \({t_0,t_f} \times R\) . Verifying the latter condition
is a great challenge since the graph of \(E\)  is not, in general,
convex. The issue has so far been resolved only for the time-invariant
special case of REF ; see [2]}.
Moreover, while ellipsoidal techniques have been generalized to handle
nonlinear dynamics, the extensions still suffer from both the
aforementioned shortcomings,
e.g. [3]}.
